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docs(AstrodynamicalSolvers): add usage examples (#266)
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‎lib/AstrodynamicalSolvers/docs/src/_quarto.yml‎

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author:
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name: "Joey Carpinelli"
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email: "[email protected]"
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date: "2025-03-02"
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date: "2025-08-17"
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chapters:
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- index.md
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- examples/index.qmd
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- api/index.qmd
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- part: examples/index.qmd
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chapters:
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- examples/propagation/index.qmd
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- examples/periodic-orbits/index.qmd
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- examples/manifolds/index.qmd
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- examples/plotting/index.qmd
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- part: api/index.qmd
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navbar:
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background: primary
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toc-title: "Table of Contents"
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engine: julia
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execute:
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echo: false
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echo: true
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output: true
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cache: false
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freeze: false

‎lib/AstrodynamicalSolvers/docs/src/examples/index.qmd‎

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# Examples
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_Usage examples._
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## Periodic Orbits
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This package contains differential correctors, and helpful wrapper functions, for
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finding periodic orbits within Circular Restricted Three Body Problem dynamics.
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### `Plots.jl`
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```{julia}
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#| echo: true
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using AstrodynamicalSolvers
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using AstrodynamicalModels
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using OrdinaryDiffEq
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using Plots
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μ = 0.012150584395829193
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planar = let
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ic = halo(μ, 1) # lyapunov (planar) orbit
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u = Vector(CartesianState(ic))
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problem = ODEProblem(CR3BFunction(), u, (0, ic.Δt), (μ,))
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solution = solve(problem, Vern9(), reltol=1e-14, abstol=1e-14)
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Plots.plot(solution, idxs=(:x,:y,:z), title = "Lyapunov Orbit", label=:none, size=(1600,900), dpi=400, aspect_ratio=1)
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end
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extraplanar = let
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ic = halo(μ, 2; amplitude=0.01) # halo (non-planar) orbit
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u = Vector(CartesianState(ic))
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problem = ODEProblem(CR3BFunction(), u, (0, ic.Δt), (μ,))
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solution = solve(problem, Vern9(), reltol=1e-14, abstol=1e-14)
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Plots.plot(solution, idxs=(:x,:y,:z), title = "Halo Orbit", label=:none, size=(1600,900), dpi=400, aspect_ratio=1)
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end
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Plots.plot(planar, extraplanar, layout=(1,2))
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```
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### `Makie.jl`
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```{julia}
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#| echo: true
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using AstrodynamicalSolvers
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using AstrodynamicalModels
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using OrdinaryDiffEq
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using CairoMakie
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μ = 0.012150584395829193
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sol_planar = let
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ic = halo(μ, 1) # lyapunov (planar) orbit
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u = Vector(CartesianState(ic))
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problem = ODEProblem(CR3BFunction(), u, (0, ic.Δt), (μ,))
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solution = solve(problem, Vern9(), reltol=1e-14, abstol=1e-14)
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end
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sol_extraplanar = let
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ic = halo(μ, 2; amplitude=0.01) # halo (non-planar) orbit
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u = Vector(CartesianState(ic))
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problem = ODEProblem(CR3BFunction(), u, (0, ic.Δt), (μ,))
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solution = solve(problem, Vern9(), reltol=1e-14, abstol=1e-14)
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end
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fig = CairoMakie.Figure(size=(800, 400); fontsize=11)
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ax_kwargs_common = (; aspect=:equal, azimuth=-π/3)
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ax_left = CairoMakie.Axis3(fig[1, 1];
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title = "Lyapunov Orbit",
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limits = (0.78, 0.90, -0.09, 0.09, -0.02, 1.04),
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ax_kwargs_common...,
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)
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ax_right = CairoMakie.Axis3(fig[1, 2];
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title = "Halo Orbit",
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limits = (1.05, 1.26, -0.1, 0.1, -0.02, 0.01),
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protrusions = (30, 100, 0, 0),
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ax_kwargs_common...,
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)
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CairoMakie.plot!(ax_left, sol_planar; idxs=(:x, :y, :z))
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CairoMakie.plot!(ax_right, sol_extraplanar; idxs=(:x, :y, :z))
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fig
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```
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## Manifold Computations
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Manifold computations, provided by `AstrodynamicalCalculations.jl`, can perturb
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halo orbits onto their unstable or stable manifolds.
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### `Plots.jl`
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```{julia}
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#| echo: true
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using AstrodynamicalSolvers
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using AstrodynamicalCalculations
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using AstrodynamicalModels
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using OrdinaryDiffEq
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using LinearAlgebra
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using Plots
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μ = 0.012150584395829193
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unstable = let
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ic = halo(μ, 1; amplitude=0.005)
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u = CartesianState(ic)
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Φ = monodromy(u, μ, ic.Δt, CR3BFunction(stm=true))
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ics = let
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problem = ODEProblem(CR3BFunction(stm=true), vcat(u, vec(I(6))), (0, ic.Δt), (μ,))
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solution = solve(problem, Vern9(), reltol=1e-12, abstol=1e-12, saveat=(ic.Δt / 10))
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solution.u
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end
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perturbations = [
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diverge(ic[1:6], reshape(ic[7:end], 6, 6), Φ; eps=-1e-7)
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for ic in ics
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]
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problem = EnsembleProblem(
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ODEProblem(CR3BFunction(), u, (0.0, 2 * ic.Δt), (μ,)),
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prob_func=(prob, i, repeat) -> remake(prob; u0=perturbations[i]),
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)
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solution = solve(problem, Vern9(), trajectories=length(perturbations), reltol=1e-14, abstol=1e-14)
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end
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stable = let
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ic = halo(μ, 2; amplitude=0.005)
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u = CartesianState(ic)
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Φ = monodromy(u, μ, ic.Δt, CR3BFunction(stm=true))
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ics = let
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problem = ODEProblem(CR3BFunction(stm=true), vcat(u, vec(I(6))), (0, ic.Δt), (μ,))
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solution = solve(problem, Vern9(), reltol=1e-12, abstol=1e-12, saveat=(ic.Δt / 10))
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solution.u
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end
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perturbations = [
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converge(ic[1:6], reshape(ic[7:end], 6, 6), Φ; eps=1e-7)
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for ic in ics
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]
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problem = EnsembleProblem(
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ODEProblem(CR3BFunction(), u, (0.0, -2.1 * ic.Δt), (μ,)),
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prob_func=(prob, i, repeat) -> remake(prob; u0=perturbations[i]),
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)
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solution = solve(problem, Vern9(), trajectories=length(perturbations), reltol=1e-14, abstol=1e-14)
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end
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figure = Plots.plot(;
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aspect_ratio = 1.0,
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background = :transparent,
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grid = true,
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title = "Unstable and Stable Invariant Manifolds",
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)
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Plots.plot!(figure, unstable, idxs=(:x, :y), aspect_ratio=1, label=:none, palette=:blues)
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Plots.plot!(figure, stable, idxs=(:x, :y), aspect_ratio=1, label=:none, palette=:blues)
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Plots.scatter!(figure, [1-μ], [0], label="Moon", xlabel="X (Earth-Moon Distance)", ylabel="Y (Earth-Moon Distance)", marker=:x, color=:black, markersize=10,)
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figure # hide
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```
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## `Makie.jl`
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```{julia}
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#| echo: true
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using AstrodynamicalSolvers
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using AstrodynamicalCalculations
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using AstrodynamicalModels
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using OrdinaryDiffEq
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using LinearAlgebra
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using CairoMakie
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μ = 0.012150584395829193
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unstable = let
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ic = halo(μ, 1; amplitude=0.005)
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u = CartesianState(ic)
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Φ = monodromy(u, μ, ic.Δt, CR3BFunction(stm=true))
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ics = let
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problem = ODEProblem(CR3BFunction(stm=true), vcat(u, vec(I(6))), (0, ic.Δt), (μ,))
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solution = solve(problem, Vern9(), reltol=1e-12, abstol=1e-12, saveat=(ic.Δt / 10))
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solution.u
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end
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perturbations = [
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diverge(ic[1:6], reshape(ic[7:end], 6, 6), Φ; eps=-1e-7)
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for ic in ics
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]
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problem = EnsembleProblem(
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ODEProblem(CR3BFunction(), u, (0.0, 2 * ic.Δt), (μ,)),
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prob_func=(prob, i, repeat) -> remake(prob; u0=perturbations[i]),
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)
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solution = solve(problem, Vern9(), trajectories=length(perturbations), reltol=1e-14, abstol=1e-14)
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end
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stable = let
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ic = halo(μ, 2; amplitude=0.005)
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u = CartesianState(ic)
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Φ = monodromy(u, μ, ic.Δt, CR3BFunction(stm=true))
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ics = let
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problem = ODEProblem(CR3BFunction(stm=true), vcat(u, vec(I(6))), (0, ic.Δt), (μ,))
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solution = solve(problem, Vern9(), reltol=1e-12, abstol=1e-12, saveat=(ic.Δt / 10))
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solution.u
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end
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perturbations = [
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converge(ic[1:6], reshape(ic[7:end], 6, 6), Φ; eps=1e-7)
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for ic in ics
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]
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problem = EnsembleProblem(
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ODEProblem(CR3BFunction(), u, (0.0, -2.1 * ic.Δt), (μ,)),
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prob_func=(prob, i, repeat) -> remake(prob; u0=perturbations[i]),
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)
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solution = solve(problem, Vern9(), trajectories=length(perturbations), reltol=1e-14, abstol=1e-14)
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end
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fig = CairoMakie.Figure(size=(800, 400), fontsize=20)
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ax = CairoMakie.Axis(fig[1, 1];
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xreversed = true,
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xticks = LinearTicks(5),
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yticks = LinearTicks(5),
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aspect = DataAspect(),
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xlabel = "X (Earth-Moon Distance)",
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ylabel = "Y (Earth-Moon Distance)",
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title = "Unstable and Stable Invariant Manifolds",
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titlesize = 24,
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)
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idxs = (:x, :y)
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# TODO: replace this manual workaround when
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# https://github.com/SciML/SciMLBase.jl/issues/697#issuecomment-2135801331
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# is addressed
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for (traj, color) in zip(unstable, resample_cmap(:blues, length(unstable)))
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CairoMakie.plot!(ax, traj; idxs, color)
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end
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for (traj, color) in zip(stable, resample_cmap(:blues, length(stable)))
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CairoMakie.plot!(ax, traj; idxs, color)
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end
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CairoMakie.scatter!(ax, [1-μ], [0]; marker='⨯', color=:black, markersize=50, label="Moon")
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fig
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```

‎lib/AstrodynamicalSolvers/docs/src/examples/manifolds/index.qmd‎

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